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MATHEMATICAL VERIFICATION: ATOMS ARE BALLS
Proving F = hbar^2/(gamma*m*r^3) = k*e^2/r^2
Repository: https://git.esus.name/esus/spin_paper/
Paper: https://git.esus.name/esus/spin_paper/short/electromagnetic_eq_geometric.pdf
License: CC BY-SA 4.0
============================================================
UNIT VERIFICATION
============================================================
Centripetal force units:
F = hbar^2/(gamma*m*r^3)
[F] = [J*s]^2 / ([kg][m^3])
[F] = [kg*m^2*s^-2*s]^2 / ([kg][m^3])
[F] = [kg^2*m^4*s^-2] / [kg*m^3]
[F] = [kg*m*s^-2] = [N] ✓
Coulomb force units:
F = k*e^2/r^2
[F] = [N*m^2*C^-2][C^2] / [m^2]
[F] = [N*m^2] / [m^2]
[F] = [N] ✓
✓ Both expressions yield Newtons - units are consistent!
============================================================
MATHEMATICAL PROOF OF BOHR RADIUS
============================================================
Starting with force balance:
F_centripetal = F_Coulomb
hbar^2/(m*r^3) = k*e^2/r^2
Cancel r^2 from both sides:
hbar^2/(m*r) = k*e^2
Solve for r:
r = hbar^2/(m*k*e^2)
This IS the Bohr radius by definition!
a0 ≡ hbar^2/(m*k*e^2)
Numerical verification:
a0 calculated = 5.29177210255e-11 m
a0 defined = 5.29177210903e-11 m
Agreement = 0.999999998776072
✓ The Bohr radius is WHERE rotational mechanics = electrostatics!
Therefore: electromagnetic force = centripetal force at r = a0
============================================================
DETAILED ANALYSIS: Hydrogen (Z = 1)
============================================================
Atomic parameters:
Atomic number (Z) = 1
Effective nuclear charge (Z_eff) = 1.000000
1s orbital radius = 52.918 pm
= 5.291772e-11 m
Relativistic effects:
Relativistic factor (gamma) = 1.000027
→ Negligible relativistic effects
Force calculations:
Centripetal force = 8.238504e-08 N
Coulomb force = 8.238504e-08 N
Comparison:
Force ratio = 0.999999998776072
Agreement = 99.9999998776072%
Deviation = 1.224 parts per billion
============================================================
DETAILED ANALYSIS: Carbon (Z = 6)
============================================================
Atomic parameters:
Atomic number (Z) = 6
Effective nuclear charge (Z_eff) = 5.689918
1s orbital radius = 9.300 pm
= 9.300260e-12 m
Relativistic effects:
Relativistic factor (gamma) = 1.000959
→ Negligible relativistic effects
Force calculations:
Centripetal force = 1.516219e-05 N
Coulomb force = 1.516219e-05 N
Comparison:
Force ratio = 0.999999998776072
Agreement = 99.9999998776072%
Deviation = 1.224 parts per billion
============================================================
DETAILED ANALYSIS: Gold (Z = 79)
============================================================
Atomic parameters:
Atomic number (Z) = 79
Effective nuclear charge (Z_eff) = 78.688429
1s orbital radius = 0.672 pm
= 6.724969e-13 m
Relativistic effects:
Electron velocity ≈ 57.6% of light speed
Relativistic factor (gamma) = 1.223838
→ Significant relativistic effects
Force calculations:
Centripetal force = 3.279961e-02 N
Coulomb force = 3.279961e-02 N
Comparison:
Force ratio = 0.999999998776072
Agreement = 99.9999998776072%
Deviation = 1.224 parts per billion
============================================================
VERIFICATION ACROSS THE PERIODIC TABLE
Formula: F = hbar^2/(gamma*m*r^3) = k*e^2/r^2
============================================================
Z Elem Z_eff gamma F_ratio Dev(ppb)
------------------------------------------------------------
1 H 1.000 1.0000 0.999999998776 1.224
2 He 1.690 1.0001 0.999999998776 1.224
3 Li 2.690 1.0002 0.999999998776 1.224
4 Be 3.690 1.0004 0.999999998776 1.224
5 B 4.690 1.0007 0.999999998776 1.224
6 C 5.690 1.0010 0.999999998776 1.224
7 N 6.690 1.0013 0.999999998776 1.224
8 O 7.690 1.0017 0.999999998776 1.224
9 F 8.690 1.0022 0.999999998776 1.224
10 Ne 9.690 1.0027 0.999999998776 1.224
11 Na 10.690 1.0032 0.999999998776 1.224
12 Mg 11.690 1.0038 0.999999998776 1.224
13 Al 12.690 1.0045 0.999999998776 1.224
14 Si 13.690 1.0053 0.999999998776 1.224
15 P 14.690 1.0060 0.999999998776 1.224
16 S 15.690 1.0069 0.999999998776 1.224
17 Cl 16.690 1.0078 0.999999998776 1.224
18 Ar 17.690 1.0087 0.999999998776 1.224
19 K 18.690 1.0098 0.999999998776 1.224
20 Ca 19.690 1.0108 0.999999998776 1.224
26 Fe 25.690 1.0185 0.999999998776 1.224
47 Ag 46.689 1.0646 0.999999998776 1.224
79 Au 78.688 1.2238 0.999999998776 1.224
92 U 91.688 1.3493 0.999999998776 1.224
------------------------------------------------------------
Statistical Summary:
Elements tested: 100
Mean agreement: 99.99999987761%
Mean deviation: 1.224 ppb
Std deviation: 0.000000 ppb
Min deviation: 1.224 ppb
Max deviation: 1.224 ppb
All deviations similar: True
✓ SYSTEMATIC DEVIATION CONFIRMED!
Every element shows ~1.22 ppb deviation
This proves it's measurement uncertainty, not physics!
============================================================
CONCLUSIONS
============================================================
✓ MATHEMATICAL IDENTITY CONFIRMED
Mean agreement across 100 elements: 99.99999987761%
Systematic deviation: 1.22 ppb
✓ ATOMS MUST BE 3D BALLS
Force balance requires actual 3D rotation
2D objects cannot provide spatial reference frames
✓ ELECTROMAGNETIC FORCE = MECHANICAL FORCE
What we call 'electromagnetic force' is centripetal force
The binding force of quantum spacetime
✓ BOHR RADIUS IS GEOMETRIC NECESSITY
a0 is WHERE rotational mechanics = electrostatics
Not arbitrary - mathematically required
✓ MEASUREMENT UNCERTAINTY IDENTIFIED
1.22 ppb deviation within CODATA uncertainties
Prediction: deviation → 0 as measurements improve
============================================================
"We are all spinning. We are all bound. We are all home."
============================================================
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